The Problem:

Wes wants to determine if employing a managed futures strategy will improve overall portfolio performance in both products.
Wes also wants to review the following concepts with regards to his funds:

• Skew - If we were to plot a histogram of returns, would the funds exhibit positive or negative skew? What are the implications of skew on the funds?
  
• Convexity - do QMOM / QVAL capture positive convexity (nonlinear positive returns in sharply positive or negative market environments)?
  
• Alpha - over long periods of time, do these funds deliver statistically significant alpha (outperformance of their benchmark)?

Additionally, Dr. Gray is also interested in various performance statistics.

We will conduct the following research:

• Backtest three popular managed futures products from different asset managers
  
• Combine both QMOM and QVAL with these three managed futures products
• Compute performance statistics, including skew, convexity, and alpha (with respective benchmarks), as well as other performance statistics to determine if adding a managed futures strategy on top of the quantitatively managed products adds value.

The chapter titled “Conclusion” will give an executive summary of the findings in this paper.

List of Funds and Data Sources

The data for this project are as follows:

• QMOM and QVAL, including their stated benchmarks (VOT and VOE, respectively)
• 3 managed futures products: ASFYX, AQMIX, RYMFX
• The 3-month risk-free rate from the Federal Reserve
• The SG Trend Index

Data was pulled from Riingo, the R package developed around the data provider Tiingo. More information in Riingo can be found here: https://cran.r-project.org/web/packages/riingo/riingo.pdf

We will also pull data from Fred, so we need a FRED API key, and we need to use the {FredR} package. More information in the FredR package can be found here: https://cran.r-project.org/web/packages/fredr/fredr.pdf

The first step is to extract the data from the relevant sources, including benchmarks as stated in the prospectuses for each fund. QMOM and QVAL listed two benchmarks for their funds; we will stick with the Vanguard managed product for simplicity.

Below is a brief description of each fund, their objective, and their benchmarks.

Initial Performance Analysis

Below, we will run an initial exploratory analysis using the {PerformanceAnalytics} package on each fund to understand all of the relevance performance data.
More info on PerformanceAnalytics: https://cran.r-project.org/web/packages/PerformanceAnalytics/PerformanceAnalytics.pdf.

After running this initial analysis, we will employ our managed futures strategy to see how performance can be improved.

First, we will explore the growth of $100 in a VAMI chart to understand the performance of each fund.
We will do this with a VAMI chart.
From the VAMI, it is apparent that QMOM had a great year during 2020. QVAL took a major hit at the beginning of the pandemic, but quickly recovered its performance. Many of the managed futures funds remain unchanged throughout the pandemic, which is exactly their purpose in the portfolio.

Let’s display the volatilities to understand the riskiness of each fund.

Volatility

QMOM and QVAL have significantly higher volatility than their benchmarks, VOT and VOE. This may be attributable to a number of factors, including the risks investors take when they purchase these actively-managed products. QMOM and QVAL hold a concentrated portfolio of 50 equities and uses a quantitatively-driven approach to select equities.

Value at Risk

Now we will build VaR measurements. VaR measures the historically-realized drawdown at some threshold, in this case, in the 99th percentile. Funds with more severe drawdowns are historically riskier.

From the chart above, we can see that QMOM and QVAL have some of the highest VaR measurements.
QVAL had an especially difficult 2020, as we have seen from the VAMI chart. Let’s build a table evaluating QVAL’s 2020 alone:

table_2020

##                   Parameter Measurement
## 1 Geometric Return for 2020      -0.15%
## 2       Volatility for 2020       6.74%
## 3          VaR for for 2020     -10.82%
## 4       Geometric Return SI       0.07%
## 5             Volatility SI      23.76%
## 6                    VaR SI      -2.13%

Across all metrics, QVAL had a very difficult 2020. Lower return, almost double the volatility, and double the VaR. This was not 2020’s star fund.

Drawdown

QVAL had one of the highest drawdowns at 54.88%, which was the result of the COVID-19 pandemic. VOE, the Vanguard mid-cap value index, had a similarly large drawdown of 45% during this time period. QMOM and the respective VOT index held their performance, mitigating a large drop in returns.

For the same reason as above, QVAL had one of the highest drawdowns (occuring in 2020). We will attempt to use a managed futures trading strategy to mitigate these huge drawdowns.

Skew

We want to measure the skewness of each fund. As a gentle reminder, a positively-skewed fund will realize frequent small losses and achieve infrequent, large gains to offset these losses. A negative-skewed fund will realize frequent small gains at the expense of large, infrequent losses.

Institutional investors (and essentially all investors) exhibit a behavioral bias: they prefer to capture small, frequent gains. We expect to see funds with negative skewness or insignificant skew.

The chart above confirms our hypothesis: most funds have responded to the institutional bias and have a negative skew, with small frequent gains and large drawdowns. AQMIX is the only fund with a positive skew, albeit not large in magnitude.

Convexity

Convexity is an interesting metric for quantitative research.
Wes Gray is interested in the quantitative implications of convexity in funds - the nonlinear change in the return of the funds relative to the benchmark, as explained in a parabolic second-order regression model.

As you will see below, we have an interesting way of measuring convexity. We build a parabolic regression model where we assume a combination of a linear relationship and a nonlinear (parabolic) relationship, plus an error term.

The model is as follows: Rp − Rf = α + β(Rb − Rf) + γ(Rb − Rf)^2 + εp

We will be using the PerformanceAnalytics::MarketTiming package to measure gamma, the second-order coefficient of the parabolic relationship.

From CRAN:

The Treynor-Mazuy model is essentially a quadratic extension of the basic CAPM. It is estimated using a multiple regression. The second term in the regression is the value of excess return squared. If the gamma coefficient in the regression is positive, then the estimated equation describes a convex upward-sloping regression “line”. The quadratic regression is:

Rp - Rf = alpha + beta(Rb -Rf) + gamma(Rb - Rf)^2 + epsilonp

gamma is a measure of the curvature of the regression line. If gamma is positive, this would indicate that the manager’s investment strategy demonstrates market timing ability.

More here: https://risk.edhec.edu/sites/risk/files/EDHEC_Working_Paper_The_alpha_of_a_market_timer_F.pdf

The HM model translates the behavior of a manager who succeeds in switching his market beta from a high level equal to βHM when the market return exceeds the risk-free rate to a low level of (βHM − γHM) otherwise. Admati et al. (1986) show that under the standard assumption of a joint normal distribution of asset returns, the TM model is consistent with a manager whose target beta varies linearly with his forecast for the expected market rate of return. In both models, a negative value of gamma induces negative market timing.

The above chart is very curious.

It appears that QMOM and QVAL have close to zero convexity, while the other funds (except for RYMFX) have some measure of convexity.

This is somewhat expected. We knew that we wanted to see if QMOM/QVAL had any measurement of convexity, and if so, how we could improve on the measurements of convexity using a managed futures fund. It is clear here that adding RYMFX funds to the quantitative products will enhance convexity and increase skew. We will see below the alpha of these funds using the CAPM model.

VOT and VOE have hugely negative convexity - this is to be expected, since they are not market-timing funds.

QMOM and QVAL had similar performance in 2016, 2017, 2018, and 2019. It was only until 2020 and 2021 that performance strongly diverged. I do not need to explain why.

Alpha of Funds relative to benchmark (using CAPM):

This chart is only incorporated because I wanted to evaluate the capital asset pricing model’s relationship with alpha. This does not measure alpha above the stated benchmark. We will discuss this further as we add the managed futures funds to the quantitative strategies. However, we can see that ASFYX adds alpha, while RYMFX has zero alpha and AQMIX detracts alpha.

It would be prudent, then, to incorporate ASFYX’s fund into QMOM and QVAL’s strategies. ASFYX accomplishes the goals Dr. Gray asked for: it adds convexity, it brings in skew, and it adds alpha relative to the CAPM. We do need to be careful before proceeding, however, because the measurement of convexity for ASFYX comes from its benchmark, which I am not 100% comfortable using that data.

We should ignore VOT and VOE for this analysis, because they are the benchmark funds. Their alpha is relative to the risk-free rate.

Correlation

The final analysis we’ll run is a large correlation table to show the relationship between each of the funds’ movements.

Nothing from the correlation table is surprising. There is a higher degree of correlation between the QMOM/QVAL products and their benchmarks, and a low degree of correlation between the quantitative funds and the managed futures funds, 3-month T-bills, and SG Trend Index.

Return

Let’s begin with our VAMI chart.

If you really wanted to, we could stop our analysis here. It doesn’t look like overlaying the managed futures strategy adds much in terms of volatility mitigation, performance enhancement, convexity, or any of the other factors we’ve considered.

Volatility

VaR

Drawdown

Skew

It does seem like the measurement of skew drew closer to zero as managed futures strategies were added to the fund.

Convexity

Convexity was brought closer to zero with the addition of AQMIX. For QVAL, convexity was worse with the addition of RYMFX. This is generally troubling, because our goal is to bring convexity to a positive value; it seems there is too much performance overlay with QVAL and RYMFX. Generally, for both QMOM and QVAL, the addition of AQMIX was positive for convexity.

Note how similar in performance QVAL and QVAL+RYMFX are. Truly, RYMFX does not provide the convexity benefits that we had hoped for.

Alpha of Funds relative to benchmark (using CAPM):

It looks like adding ASFYX to each fund dramatically increases the monthly alpha. From a CAPM standpoint, it is prudent to evaluate ASFYX further. Combining what we know about ASFYX with skew and convexity, the best course of action is to add ASFYX’s fund to QMOM.

Correlation

The final analysis we’ll run is a large correlation table to show the relationship between each of the funds’ movements.

Return

Let’s begin with our VAMI chart.

If you really wanted to, we could stop the analysis here. It doesn’t look like overlaying the managed futures strategy adds much in terms of volatility mitigation, performance enhancement, convexity, or any of the other factors we’ve considered.

Volatility

VaR

Drawdown

Skew

It does seem like the measurement of skew drew closer to zero as managed futures strategies were added to the fund.

Convexity

We come to the same point here in convexity as we’ve explored in prior convexity charts. QMOM benefits from AQMIX, and QVAL detracts in convexity with the addition of RYMFX. It does not look like that there is any reasonable measure of a managed futures strategy that bring a positive convexity to the quantitative products.

Alpha of Funds relative to benchmark (using CAPM):

The point here just enhances the discussion on alpha from earlier - ASFYX is a prudent addition to QMOM.

Correlation

The final analysis we’ll run is a large correlation table to show the relationship between each of the funds’ movements.

Yes, adding managed futures funds to each of the strategies increases the overall correlation; this is to be expected.